$A$ bomb of mass $16 \ kg$ at rest explodes into two pieces of masses $4 \ kg$ and $12 \ kg.$ The velocity of the $12 \ kg$ mass is $4 \ m s^{-1}.$ The kinetic energy of the other mass is .............. $J$.

$A$ particle of mass $m$ is moving along the $x$-axis with a speed $v_0$. Suddenly,a part of its mass,$m/3$,breaks off and moves parallel to the $y$-axis with a speed $2v_0$. Find the velocity of the remaining part in terms of unit vectors.

$A$ vessel at rest explodes into three pieces. Two pieces having equal masses $m$ fly off perpendicular to one another with the same velocity $30 \, m/s$. The third piece has three times the mass of each of the other pieces. The magnitude and direction of the velocity of the third piece will be:

$A$ man of mass $m$ is standing on a platform of mass $M$ kept on a smooth ice surface. If the man starts moving on the platform with a speed $v$ relative to the platform,with what velocity does the platform recoil relative to the ice?

The figure shows a man of mass $m$ standing at the end $A$ of a trolley of mass $M$ placed at rest on a smooth horizontal surface. The man starts moving towards the end $B$ with a velocity $u_{rel}$ with respect to the trolley. The length of the trolley is $L$. The time taken by the man to reach the other end is

$A$ shell at rest at the origin explodes into three fragments of masses $1 \ kg$,$2 \ kg$,and $m \ kg$. The $1 \ kg$ and $2 \ kg$ pieces fly off with speeds of $5 \ m/s$ along the $x$-axis and $6 \ m/s$ along the $y$-axis,respectively. If the $m \ kg$ piece flies off with a speed of $6.5 \ m/s$,the total mass of the shell must be ......... $kg$.